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Pardon if this is well known. Suppose I have a (say complex) connected reductive group $G^{\vee}$ with the $\tilde{\Delta}=\Delta\cup\{\alpha_0\}$ being the simple roots plus the negative highest root of $G^{\vee}$. Any proper subset of $\tilde{\Delta}$ defines a connected reductive subgroup $H^{\vee}\subset G^{\vee}$. Let $H$ and $G$ be the corresponding dual groups. For any special nilpotent orbit $u_H$ in $H$ we can define the Lusztig-Spaltenstein induction $u$ as a nilpotent orbit in $G$.

(that is, $(u_H,triv)$ corresponds to some representation of $W_H$ by Springer correspondence, we do the $j$-induction on this to get a representation of $W$, which correspond to $(u,triv)$ for some $u$)

Question: Is it true that the codimension of $u$ in the nilpotent cone of $G$ is the same is the codimension of $u_H$ in the nilpotent cone of $H$?

Thanks!

Edited: Just to say what I understand / guess so far: For any $u_H$, say the Springer fiber $\mathcal{B}_{e_H}$ at $u_H$ has dimension $e_H$. Then the corresponding Weyl group representation appears as the component of $H^{2e_H}(\mathcal{B}_{e_H})$ on which $A_{e_H}$ acts trivially. This component receives a surjection from $H^{2e_H}(H/B_H)$, which is a quotient of $\text{Sym}^{2e_H}(std_H)$, where $std_H$ is the standard representation of $W_H$. The Lusztig-Spaltenstein $j$-induction then, I think (I cannot find a reference for why this works for $H^{2e_H}(H/B_H)$ instead of $\text{Sym}^{2e_H}(std_H)$), gives a $W$-representation in $H^{2e_H}(G/B)$, which then corresponds to some orbit in $G$ whose Springer fiber has dimension $e_H$.

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    $\begingroup$ I don't think you want to call the highest root $\rho$ (except in $SL_3$) because that's standardly used for half the sum of the positive roots. $\endgroup$
    – Allen Knutson
    Commented Oct 7, 2015 at 1:50
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    $\begingroup$ Theorem 1.3(b) of Lusztig-Spaltenstein (Induced unipotent class) says the springer fiber of $u$ in $G$ has the same dimension as that of $u_H$ in $H$ if $H$ is a Levi of $G$. Is it going to help? $\endgroup$
    – wky
    Commented Oct 7, 2015 at 2:04
  • $\begingroup$ Definitely! I think their construction showed that the assertion is true when $H$ is a Levi (e.g. via the theorem you pointed out). $\endgroup$
    – Cheng-Chiang Tsai
    Commented Oct 7, 2015 at 3:25
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    $\begingroup$ I learned from Prof. Lusztig that this is (essentially) true, and really follows from that in the definition of $j$-induction, the minimum number $e$ so that the interested rep'n of the Weyl group appears in $\mathrm{Sym}^e(std)$ is the same as the dimension for the Springer fiber. I'll try to add reference later when I read them better. $\endgroup$
    – Cheng-Chiang Tsai
    Commented Oct 9, 2015 at 17:03
  • $\begingroup$ @Tsai: As Jay points out (and Lusztig suggests), there are relevant details in Lusztig's papers. The extra groups $H$ are called "pseudo-Levi subgroups" by his former student Eric Sommers, though Eric and I disagree about whether that label should include the genuine Levi subgroups of parabolic subgroups. $\endgroup$
    – Jim Humphreys
    Commented Oct 18, 2015 at 17:08

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What you're asking for can be phrased entirely in terms of nilpotent orbits. If $\mathcal{O}$ is the nilpotent orbit of $H$ then the nilpotent orbit of $G$ you obtain via your process is the induced nilpotent orbit $\mathrm{Ind}_H^G(\mathcal{O})$. It is well known that the codimensions of these orbits coincide. See Proposition 7.1.4 of the book "Nilpotent Orbits in Semisimple Lie Algebras" by Collingwood and McGovern.

EDIT: Sorry, I just saw that you want this for a more general class of subgroups. The statement is still true. See 13.3 in Lusztig's book "Characters of reductive Groups over a finite field" and also Lusztig's paper "Unipotent classes and special Weyl group representations", J. Algebra (321), no. 11, 3418-3449, where more details are provided concerning some of the statements in the book.

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